A264398 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with odd multiplicities.

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 1, 0, 4, 3, 3, 4, 3, 1, 0, 7, 7, 1, 5, 7, 7, 3, 0, 12, 14, 4, 7, 12, 14, 8, 1, 0, 19, 26, 10, 1, 11, 19, 26, 18, 3, 0, 30, 45, 22, 4, 15, 30, 45, 36, 9, 0, 45, 75, 44, 11, 1, 22, 45, 75, 67, 21, 1, 0, 67, 120, 81, 26, 3, 30, 67, 120, 119, 45, 4

Offset

0,7

Comments

T(n,0) = A035363(n).

Row sums give A000041. - Omar E. Pol, Nov 21 2015

Links

Alois P. Heinz, Rows n = 0..1000, flattened

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30

Formula

G.f.: G(t,x) = Product_{j>=1} (1 + tx^j)/(1 - x^(2j)).

Sum_{k>0} k * T(n,k) = A209423(n). - Alois P. Heinz, Aug 05 2020

G.f.: A(x,y)*B(x^2) where A(x),B(x) are the o.g.f.'s for A008289 and A000041. (See Flajolet, Sedgewick link.) - Geoffrey Critzer, Aug 07 2022

Example

T(6,1) = 4 because we have [6*], [4*,1,1],[2*,2,2], and [2*,1,1,1,1] (parts with odd multiplicities are marked).
Triangle starts:
  1;
  0, 1;
  1, 1;
  0, 2, 1;
  2, 2, 1;
  0, 4, 3;
  3, 4, 3, 1;
  ...

Maple

g := product((1+t*x^j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(`if`(j::odd, x, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 25 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[OddQ[j], x, 1]* b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2016, after Alois P. Heinz *)

Prog

(PARI)
T(n) = { Vec(prod(k=1, n, (1 + y*x^k)/(1 - x^(2*k)) + O(x*x^n))) }
{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

Crossrefs

Cf. A000041, A035363, A209423, A264399, A264400.

Sequence in context: A178064 A145363 A071429 * A145364 A284978 A175862

Adjacent sequences: A264395 A264396 A264397 * A264399 A264400 A264401

Keyword

nonn,look,tabf

Author

Emeric Deutsch, Nov 21 2015

Status

approved